Empirical and Analytical Correlations 307

4.4.1Viscous Dissipation Domain

One of the earliest analytical models for the calculation of flattening ratio of a droplet impinging on a solid surface was developed by Jones.[508] In this model, the effects of surface tension and solidification were ignored. Thus, the flattening ratio is only a function of the Reynolds number. Discrepancies between experimental results and the predictions by this model have been reported and discussed by Bennett and Poulikakos.[380]

In case that the decay of impact kinetic energy due to viscous dissipation is the predominant mechanism in droplet flattening, Madejski’s full model reduces to:

Eq. (49)

Ds / D0 = 1.2941(Re+ 0.9517 )0.2

This is sometimes referred to as Madejski’s flow model in literature. In a subsequent study, Madejski[520] carried out calculations using the same model for a large range of Reynolds and Weber numbers. It was indicated that the model is applicable to plasma spraying of ceramic powder for surface coating. For Al2O3 powder, for example, a flattening ratio of 5–6 can be achieved for 1000≤We≤10000 and 4000≤Pe≤8000. Other application fields include: (1) Boiler tubes at mist flow and post-critical conditions (50≤We≤300, 200≤Re≤20000);

(2) Last stages of steam turbines at mist flow conditions after condensation shock (20≤We≤50,50≤Re≤200); (3) Rain erosion of aircraft (50000≤We≤300000, 40000≤Re≤200000); (4) Aircraft icing (500≤We≤3500, 400≤Re≤1400); (5) Splat quenching of alloys (mainly aluminum alloys) (50≤We≤500, 500≤Re≤10000).

An analysis very similar to Madejski’s study[401] has been conducted by Fiedler and Naber[521] for normal liquids in combustion applications from which solidification phenomenon is absent. The derived results are almost identical to Madejski’s model. Markworth and Saunders[522] presented a substantial improvement upon the velocity field used by Madejski, and demonstrated a correspondingly improved model prediction.

308 Science and Engineering of Droplets

Trapaga and Szekely[515] conducted a detailed numerical analysis of droplet flattening process on a flat surface. A fit to the numerical results yielded a correlation similar to Madejski’s flow model, but with a slightly smaller proportional coefficient. Hamatani et al.[516] employed a numerical algorithm to model the impact of alumina droplets assuming a constant droplet temperature during deformation and investigated the effects of droplet size, velocity and viscosity on the deformation process. They also derived a correlation similar to Madejski’s flow model, but with an even smaller coefficient. A very similar correlation has been reported by Watanabe et al.[517]

Overall, if the decay of kinetic energy of an impacting droplet is due to viscous dissipation during flattening, the corresponding models in this group predict that a large flattening ratio can be obtained for large droplets with high density and/or low viscosity at high impact velocity. For large Weber numbers, the droplet flattening behavior is dependent only on the Reynolds number. A notable example is thermal spray applications, in which surface tension forces are shown to be unimportant to droplet flattening processes.[401][508]

4.4.2Surface Tension Domain

For droplets of high surface tension, the droplet flattening process may be governed by the transformation of impact kinetic energy to surface energy. In case that this mechanism dominates, the flattening ratio becomes only dependent on the Weber number, as derived by Madejski by fitting the numerical results of the full analytical model:

Eq. (50)

Ds / D0 = (We / 3)0.5

The values of the Weber number pertinent to this correlation were suggested to be sufficiently small so that viscous and solidification effects can be neglected. Another analytical expression, derived from Madejski’s full model after simplification under the conditions

Empirical and Analytical Correlations 309

that surface tension dominates the droplet flattening, has no limitation on the value of the Weber number:[390]

Eq. (51)

We = 3(Ds / D0 )2 + 8 /(Ds / D0 ) − 11

As pointed out by Dykhuizen,[390] the assumptions made in Madejski’s model in terms of surface energy may be questionable. The model ignored the surface energy associated with the liquid-solid and gassolid interfaces, and included only the liquid-gas interface. Thus, the flattening process in the model is independent of the liquid contact angle. In reality, however, the droplet flattening behavior is intimately related to the contact angle in flattening processes at low Weber numbers.[380][391] Therefore, Madejski’s model is valid only for the condition of 90° contact angle.[390] Under such special condition, the ignored two surface energy terms will vanish. In general, the contact angle is a complicated function of contact-line speed.[391] The contact line is a curve formed when an interface between two immiscible fluids (for example, gas and liquid) intersects a solid. The effect of the contact line on droplet spreading has been investigated by Haley and Miksis[523] on the basis of lubrication theory. They considered several slip coefficients and relationships between contact-line speed and contact angle, and found that the spreading rates strongly depend on the relationships, although the qualitative features of the droplet spreading are similar for the different relationships.

Cheng[509] studied dynamic spreading of droplets impacting onto a solid surface without wetting and solidification. A significantly different expression was derived for the maximum degree of flattening under the conditions where surface tension is the only retarding force to droplet spreading:

Eq. (52)

Ds / D0 = 0.816 We0.25

The correlations derived by Chandra and Avedisian,[411] and Collings et al.[514] are similar to Madejski’s equation[401] except that the

310 Science and Engineering of Droplets

former include the effect of contact angle. However, there are some problems associated with these correlations, and discrepancies between experimental results and the predictions by these correlations have been reported, as discussed in detail by Bennett and Poulikakos.[380] For example, Chandra and Avedisian’s model[411] predicted a maximum spread factor that is 20–40% higher than some experimentally measured results. This discrepancy was attributed to the deficient model for viscous energy dissipation. The model of Collings et al.[514] predicted a maximum spread factor 100% too small. The grossly large discrepancy may indicate that the treatment of the equilibrium contact angle in their model was incorrect.[380] These also demonstrated the importance of contact angle consideration and inclusion of the initial droplet surface tension in analytical models for predicting the flattening ratio.

4.4.3Solidification Domain

In Madejski’s full model,[401] solidification of melt droplets is formulated using the solution of analogous Stefan problem. Assuming a disk shape for both liquid and solid layers, the flattening ratio is derived from the numerical results of the solidification model for large Reynolds and Weber numbers:

Eq. (53)	D / D = 1.4991(ρ			L	/ ρ	S	)0.395	(Pe /U 2 )0.1975
	s	0
Dykhuizen[390] introduced a simplified model:
Eq. (54)	Ds	/ D0	= 0.82(Pe /U 2 )0.25

which compares well with Madejski’s solidification model.

To use these models, the freezing constant, U, must be determined. One choice is the solution of the Stefan problem of solidification, as described by Madejski:[401]

Empirical and Analytical Correlations

311

Eq. (55)

U

T

Tp

(kc p ρ )L /(kc p ρ )S

π =

0

−

2

é

ù

æ U

ö

æ U 2

ö

æ U 2

ö

aS

aS

U

(kc

ρ)

êerf

+

p

S

ú expç

÷

erfcç

÷ expç

÷

2 a

a

ê

2

(kcp ρ )

ú

ç

÷

ç

L

÷

ç 4

÷

è 4

ø

è

ø

è

L ø

ë

W

û

where a = k/cpρ is the thermal diffusivity, T0 = kS(Tm - Tw)/(aSρSDHm) and Tp = kS(TL – Tm)/(aSρSDHm) are the dimensionless substrate temperature and dimensionless droplet temperature, respectively. Subscripts L, S, and W denote liquid droplet, solidified material, and substrate, respectively. This equation is derived for 1-D conductionlimited solidification of a molten layer at an initial temperature TL that comes in contact with a substrate at an initial temperature Tw. The equation dictates that splat solidification is dependent on substrate thermal properties, conflicting with many experimental data. Thus, Madejski assumed that the substrate remained isothermal during the cooling process that requires the freezing constant be calculated from the following equation:

		T0	=			U	erf	U	exp(U 2 / 4)
	Eq. (56)			π
			2				2

To rationalize the isothermal assumption, Dykhuizen[390] discussed two related physical phenomena. First, heat may be drawn out of the substrate from an area that is much larger than that covered by a splat. Thus, the 1-D assumption in the Stefan problem becomes invalid, and a solution of multidimensional heat conduction may make the interface between a splat and substrate closer to isothermal. Second, the contact resistance at the interface is deemed to be the largest thermal resistance retarding heat removal from the splat. If this resistance does not vary much with substrate material, splat solidification should be independent of substrate thermal properties. Either of the phenomena would result in a heat-transfer rate that is less dependent on the substrate properties, but not as high as that calculated by Madejski based on the

312 Science and Engineering of Droplets

isothermal assumption. Thus, for most thermal spray conditions, Madejski’s solidification model predicts a splat that is smaller than those predicted by his flow and surface-tension models. This suggested that solidification process acted faster than viscous dissipation process, and hence, were the predominant mechanism in limiting splat flattening. On the contrary, Jones[508] predicted a solidification rate that is significantly lower due to thermal contact resistance, dismissing the solidification mechanism in favor of viscous dissipation mechanism. Jones’ scaling argument is often quoted for thermal spray conditions where the existence of a contact resistance is much more important than in millimeter-sized droplet experiments. The contact resistance has been calculated by Moreau et al.[524] on the order of 10-6 Km2/W for plasma-sprayed molybdenum on various substrates, and measured by Fantassi et al.[406] as2×10-6 Km2/W, which are in general agreement with the values (10-4 to 10-6 Km2/W) assumed by Clyne,[155] and (2×10-5 Km2/W) by Jones.[508]

In addition, Madejski’s solidification model did not account for any time delay for nucleation of solid phase due to contact resistance and undercooling of liquid phase. The nucleation delay due to undercooling is common in rapid solidification processes, and may be an order of magnitude longer than impact time.[155] Nucleation delay becomes increasingly important as the droplet size decreases and the impact velocity increases. Thus, Dykhuizen[390] suggested to introduce an additional dimensionless number to determine when contact resistance and nucle-

ation delay are important. Dykhuizen further proposed this number be

· ≈ formulated as a ratio of nucleation time ( T + 0.5 Hm/cpL)/T (0.5 Hm/

·

cpL)/T to impact time D0/u0 (Table 4.22). In the present book, this number will be referred to as Nucleation number. The Nucleation number is identical to the Freezing number proposed by Matson et al.[409] per se. If the value of this number is small, nucleation delay needs not to be considered. In millimeter-sized droplet experiments, the Nucleation number is small, but it is five orders of magnitude larger inthermal spray applications[390] where micrometer-sized droplets impact substrate at high speeds. Hence, extrapolation of experimental results for millimeter-sized droplets to thermal spray applications may not be easily justified.